#9716127

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,8)(3,7,10,4)(5,13,15,9)(11,12,16,14)(17,18)(19,29)(20,23)(21,26)(22,27)(24,28)(25,31)(30,32)(36,44)(39,46)(42,48)(45,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19,24,31)(18,25,28,29)(20,30,21,22)(23,27,26,32)(33,34)(35,40)(36,39)(37,41)(38,43)(42,47)(44,46)(45,48);(2,8)(3,10)(9,13)(11,16)(17,20,24,21)(18,23,28,26)(19,30,31,22)(25,27,29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,14)(15,16)(17,21,24,20)(18,26,28,23)(19,22,31,30)(25,32,29,27)(33,36)(34,39)(35,42)(37,44)(38,45)(40,47)(41,46)(43,48);(1,4)(2,3)(5,14)(6,7)(8,10)(9,16)(11,13)(12,15)(17,22)(18,27)(19,20)(21,31)(23,29)(24,30)(25,26)(28,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5,6,15)(2,9,8,13)(3,11,10,16)(4,12,7,14)(17,23,24,26)(18,20,28,21)(19,27,31,32)(22,25,30,29)(33,34)(35,38)(36,39)(37,41)(40,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9623323);